`minres`

minimum residual algorithm

# Synopsis

template <class Matrix, class Vector, class Preconditioner>
int minres (const Matrix &A, Vector &x, const Vector &Mb, const Preconditioner &M,
const solver_option& sopt = solver_option())
rheolef::minres
int minres(const Matrix &A, Vector &x, const Vector &Mb, const Preconditioner &M, const solver_option &sopt=solver_option())
Definition: minres.h:100
M
Expr1::memory_type M
Definition: vec_expr_v2.h:416

# Description

This function solves the symmetric positive but possibly singular linear system `A*x=b` with the minimal residual method. The `minres` function follows the algorithm described in

        C. C. Paige and M. A. Saunders,
        Solution of sparse indefinite systems of linear equations",
        SIAM J. Numer. Anal., 12(4), 1975.

For more, see http://www.stanford.edu/group/SOL/software.html and also at page 60 of the PhD report:

       S.-C. T. Choi,
       Iterative methods for singular linear equations and least-squares problems,
       Stanford University, 2006,
       http://www.stanford.edu/group/SOL/dissertations/sou-cheng-choi-thesis.pdf

# Example

        solver_option sopt;
        sopt.max_iter = 100;
        sopt.tol = 1e-7;
        int status = minres (A, x, b, eye(), sopt);

The fourth argument of `minres` is a preconditionner: here, the `eye` one is a do-nothing preconditionner, for simplicity. Finally, the `solver_option` variable `sopt` transmits the stopping criterion with `sopt.tol` and `sopt.max_iter`.

On return, the `sopt.residue` and `sopt.n_iter` indicate the reached residue and the number of iterations effectively performed. The return `status` is zero when the prescribed tolerance `tol` has been obtained, and non-zero otherwise. Also, the `x` variable contains the approximate solution. See also the `solver_option` for more controls upon the stopping criterion.

# Implementation

This documentation has been generated from file linalg/lib/minres.h

The present template implementation is inspired from the `IML++ 1.2` iterative method library, http://math.nist.gov/iml++

template <class Matrix, class Vector, class Preconditioner>
int minres (const Matrix &A, Vector &x, const Vector &Mb, const Preconditioner &M,
const solver_option& sopt = solver_option())
{
// Size &max_iter, Real &tol, odiststream *p_derr = 0
typedef typename Vector::size_type Size;
typedef typename Vector::float_type Real;
std::string label = (sopt.label != "" ? sopt.label : "minres");
Vector b = M.solve(Mb);
Real norm_b = sqrt(fabs(dot(Mb,b)));
if (sopt.absolute_stopping || norm_b == Real(0.)) norm_b = 1;
Vector Mr = Mb - A*x;
Vector z = M.solve(Mr);
Real beta2 = dot(Mr, z);
Real norm_r = sqrt(fabs(beta2));
sopt.residue = norm_r/norm_b;
if (sopt.p_err) (*sopt.p_err) << "[" << label << "] #iteration residue" << std::endl
<< "[" << label << "] 0 " << sopt.residue << std::endl;
if (beta2 < 0 || sopt.residue <= sopt.tol) {
dis_warning_macro ("beta2 = " << beta2 << " < 0: stop");
return 0;
}
Real beta = sqrt(beta2);
Real eta = beta;
Vector Mv = Mr/beta;
Vector u = z/beta;
Real c_old = 1.;
Real s_old = 0.;
Real c = 1.;
Real s = 0.;
Vector u_old (x.ownership(), 0.);
Vector Mv_old (x.ownership(), 0.);
Vector w (x.ownership(), 0.);
Vector w_old (x.ownership(), 0.);
Vector w_old2 (x.ownership(), 0.);
for (sopt.n_iter = 1; sopt.n_iter <= sopt.max_iter; sopt.n_iter++) {
// Lanczos
Mr = A*u;
z = M.solve(Mr);
Real alpha = dot(Mr, u);
Mr = Mr - alpha*Mv - beta*Mv_old;
z = z - alpha*u - beta*u_old;
beta2 = dot(Mr, z);
if (beta2 < 0) {
dis_warning_macro ("minres: machine precision problem");
sopt.residue = norm_r/norm_b;
return 2;
}
Real beta_old = beta;
beta = sqrt(beta2);
// QR factorisation
Real c_old2 = c_old;
Real s_old2 = s_old;
c_old = c;
s_old = s;
Real rho0 = c_old*alpha - c_old2*s_old*beta_old;
Real rho2 = s_old*alpha + c_old2*c_old*beta_old;
Real rho1 = sqrt(sqr(rho0) + sqr(beta));
Real rho3 = s_old2 * beta_old;
// Givens rotation
c = rho0 / rho1;
s = beta / rho1;
// update
w_old2 = w_old;
w_old = w;
w = (u - rho2*w_old - rho3*w_old2)/rho1;
x += c*eta*w;
eta = -s*eta;
Mv_old = Mv;
u_old = u;
Mv = Mr/beta;
u = z/beta;
// check residue
norm_r *= s;
sopt.residue = norm_r/norm_b;
if (sopt.p_err) (*sopt.p_err) << "[" << label << "] " << sopt.n_iter << " " << sopt.residue << std::endl;
if (sopt.residue <= sopt.tol) return 0;
}
return 1;
}
dis_warning_macro
#define dis_warning_macro(message)
Definition: dis_macros.h:66
bdf::alpha
Float alpha[pmax+1][pmax+1]
Definition: bdf.icc:28
mkgeo_ball.b
int b
Definition: mkgeo_ball.sh:152
mkgeo_ball.c
int c
Definition: mkgeo_ball.sh:153
rheolef::dot
rheolef::std enable_if ::type dot const Expr1 expr1, const Expr2 expr2 dot(const Expr1 &expr1, const Expr2 &expr2)
dot(x,y): see the expression page for the full documentation
Definition: vec_expr_v2.h:415
rk::beta
Float beta[][pmax+1]
Definition: runge_kutta_semiimplicit.icc:60
eta
Definition: eta.h:25
rheolef::Vector::size_type
DATA::size_type size_type
Definition: Vector.h:188
u
Definition: leveque.h:25
u
Float u(const point &x)
Definition: transmission_error.cc:26